# Frank-Wolfe Algorithm for the Exact Sparse Problem

## TL;DR

This paper analyzes the Frank-Wolfe algorithm's effectiveness for the Exact Sparse reconstruction problem, showing it efficiently identifies support atoms and converges exponentially fast under quasi-incoherent dictionaries.

## Contribution

It provides theoretical guarantees for support recovery and convergence speed of the Frank-Wolfe algorithm in sparse reconstruction with quasi-incoherent dictionaries.

## Key findings

- Supports are correctly identified at each iteration.
- Exponential convergence is achieved beyond a certain iteration.
- Algorithm performance depends on dictionary incoherence.

## Abstract

In this paper, we study the properties of the Frank-Wolfe algorithm to solve the \ExactSparse reconstruction problem. We prove that when the dictionary is quasi-incoherent, at each iteration, the Frank-Wolfe algorithm picks up an atom indexed by the support. We also prove that when the dictionary is quasi-incoherent, there exists an iteration beyond which the algorithm converges exponentially fast.